Paul and Silas were Roman citizens, so it was illegal to beat and imprison them (16:22–24). When the city officials learned this, they were understandably alarmed because they had broken Roman law.
Let them come themselves and escort us out!: Paul used his rights as a Roman citizen to save his reputation and protect the Christian message from receiving a bad name. Luke wanted to show that Christianity should have the same status as Judaism in the Roman Empire as a legal religion.
The Christian faith was not at odds with Roman citizenship. It was important to note that Paul used his rights as a Roman citizen. He was willing to use his privileges as a citizen to promote Christianity in a hostile world (comparing 22:25–27).